Applications of the remainder theorem and the Rational roots theorem.

1. Rational Roots Theorem
(really this time)
At the Feet of an Ancient Master
by flickr user premasagar

4. Determine each value of k.
(a) When x 3 + kx2 + 2x - 3 is divided by x + 2, the remainder is 1.

5. Determine each value of k.
(a) When x 3 + kx2 + 2x - 3 is divided by x + 2, the remainder is 1.

6. Determine each value of k.
(b) When x4 - kx 3 + 2x2 + x + 4 is divided by x - 3, the remainder is 16.

7. When the polynomial 2x 2 + bx - 5 is divided by x - 3, the remainder is 7.
(a) Determine the value of b.
(b) What is the remainder when the polynomial is divided by x - 2?

8. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
Step 1: Find all possible ƒ(x) = 3x3 - 4x2 - 5x + 2
numerators by listing the
positive and negative
1, -1, 2, -2
factors of the constant
term.

9. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
ƒ(x) = 3x3 - 4x2 - 5x + 2
Step 2: Find all possible
denominators by listing
the positive factors of the 1, 3
leading coefficient.

10. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
ƒ(x) = 3x3 - 4x2 - 5x + 2
Step 3: List all possible
rational roots. Eliminate
1, -1, 2, -2
all duplicates.
1, 3

11. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
ƒ(x) = 3x3 - 4x2 - 5x + 2
Step 4: Use synthetic division and
the factor theorem to reduce ƒ(x)
to a quadratic. (In our example,
we’ll only need one such root.)
is a root!
-1
So,

12. Rational Roots Theorem
For any polynomial function
if P(x) has rational roots, they may be found using this procedure:
Example
Procedure
Step 5: Factor the quadratic.
Step 6: Find all roots.

13. Rational Roots Theorem
You try ...
ƒ(x) = x3 + 3x 2 - 13x - 15